Conductor of an order (awaiting review)

The conductor $\mathfrak{f}_R$ of an order $R$ in an étale algebra $K$ over $\Q$ with maximal order $\mathcal{O}_K$ is the biggest $\mathcal{O}_K$-ideal contained in $R$, that is, \[ \mathfrak{f}_R = (R:\mathcal{O}_K) = \{ x \in K :\ x\mathcal{O}_K \subseteq R \}. \]

If $R\subseteq S$ the (relative) conductor of $R$ in $S$ is the biggest $S$-ideal contained in $R$, that is, \[ (R:S) = \{ x \in K :\ xS \subseteq R \}. \]